# reference request – The Guinand-Weil categorical formulation for Hecke characters Answer

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reference request – The Guinand-Weil categorical formulation for Hecke characters

The Guinand-Weil formulation for the Riemann zeta operate is
commence{aligned}&Phi (1)+Phi (0)-sum _{rho }Phi (rho )={}&sum _{p,m}{frac {log(p)}{p^{m/2}}}{Big (}F(log(p^{m}))+F(-log(p^{m})){Big )}-{frac {1}{2pi }}int _{-infty }^{infty }varphi (t)Psi (t),dtend{aligned}

the place $$rho$$ runs over the nontrivial zeros of the Riemann zeta operate, $$p$$ runs over the primes, $$m$$ runs over the constructive integers, $$F$$ is a quickly lowering operate, and $$Phi(1/2+it)=varphi(t)$$ is the Fourier remodel of $$F$$.

To quote Wikipedia:

More usually, the Riemann zeta operate and the L-series can breathe changed by the Dedekind zeta operate of an algebraic quantity bailiwick or a Hecke L-series. The sum over primes then will get changed by a sum over prime beliefs.

Q: What is the Guinand-Weil formulation for Hecke L-functions, the place there’s a sum over Hecke L-function zeros of the Fourier remodel of a quickly lowering operate on the LHS, and a summation over prime beliefs on the RHS?

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